The lattice studies offer interesting problems concerning the relation between the size of chiral condensates and the chiral restoration temperature.
Magnetic fields enhance the size of chiral condensates (magnetic catalysis [1]}, [2]}, [3]}) but reduce the melting (chiral restoration) temperature [4]}.
The latter is called the inverse magnetic catalysis.
This phenomenon is in conflict with
the intuition that a larger chiral condensate leads to a larger dynamical quark mass as well as a larger transition temperature.
Such intuition is largely based on the experiences from the studies of the Nambu-Jona-Lasinio (NJL) type models with the fixed couplings;
at finite \(B\) , the NJL models lead to significant enhancement in the effective quark masses, chiral condensates, and the restoration temperatures, see, e.g., Refs. [5]}, [6]}
for early works and recent reviews [7]}, [8]}.
To cure the problem,
seminal works proposed the \(B\) -dependent four Fermi couplings in the NJL models [9]}, [10]}, [11]}, [12]}
or studied higher order effects such as meson fluctuations or quark-meson couplings [13]}, [14]}, [15]}, [16]}.
We note that, in models without the dynamical quark mass generation, the inverse magnetic catalysis is not necessarily a paradoxical phenomenon;
indeed, for a fixed quark mass, several finite temperature calculations with quarks predict the reduction of the critical temperature, see e.g., Refs.[17]}, [18]}.
